‘Don’t’ is a contraction of ‘do not.’ When using ‘why’ you are asking a question that involves yourself and the person you are talking to.
Here are some examples:
“Why don’t we go bowling tonight?”
“Why don’t we pick some fresh flowers?”
“Why don’t we play a game of chess?”
“Why don’t we save more money?”
“Why don’t we remember this place?”
“Why don’t we test this before using it?”
“Why don’t we try and do it again?”
“Why don’t we post our results online?”
“Why don’t we gather more firewood?”
“Why don’t we earn more money?“